Robbie Neilson's side won 4-0 on aggregate in front of 8,540 fans at Tannadice and will now face St Mirren or Hamilton Accies next Thursday and Sunday, with the first leg at home.
ICT had the better of the first half but fell behind to a Nicky Clark penalty and Osman Sow added the second goal nine minutes after the interval before Pavol Safranko wrapped things up.
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